Singular values and evenness symmetry in random matrix theory

Abstract

Complex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two independent eigenvalue sequences distributed according to particular matrix ensembles with chiral unitary symmetry. We give decompositions of the distribution of singular values, and the decimation of the singular values - whereby only even, or odd, labels are observed - for real symmetric random matrices with an orthogonal symmetry, and even weight. This requires further specifying the functional form of the weight to one of three types - Gauss, symmetric Jacobi ..